Measuring our Universe

Transcription

1 Measuring our Universe Larry P. Ammann A remarkable relation between the brightness of these variables and the length of their periods will be noticed. mathematics, now called trigonometry, to observations of a solar eclipse at two different sites to estimate the Earth-Moon distance in terms of the radius of the Earth. Figure 1 illustrates a simplified version of the geometry of these observations. It assumes that the Eclipse occurred at noon, the two sites were on a north-south line, and the Sun is infinitely distant. Correcting for these details won t have much impact on the final answer. These simple words, written nearly 100 years ago, heralded a discovery that profoundly changed mankind s view of our Universe its size, its beginning, its composition, and its future. To understand properly the importance of this discovery and the extraordinary person responsible for it, we first must review how distances in our Universe were measured prior to the beginning of the 20 th century. Figure 1: Geometry of a Solar Eclipse from Two Locations At position A (Hellespont, Greece) the Moon completely eclipsed the Sun, but at position B (Alexandria, Egypt), the Moon only partially eclipsed the Sun, leaving approximately 1/5 of the Sun s diameter uncovered. Hipparchus knew the latitudes of these two sites (his values were 41 at A and 31 at B) and he also knew that the full diameter of the Sun covered approximately 1/650 (0.554 ) of its circular path around the sky. This implies that EDF = CDB = Since AB is the base of an isosceles triangle with vertex C, then CAB = 85. At the time of this eclipse, the Sun and Moon were 3 below the Celestial Equator, and so the angle between the Moon and the Zenith at A would have been = 44. Therefore, BAD = = 51. We now can apply the law of sines to the oblique triangle ABD, Solar System Distances: Earth and Moon In the 3 rd century BCE, the Greek mathematician Eratosthenes was able to estimate the Earth s circumference. He learned that on the summer solstice the Sun at noon was reflected in a deep well in the southern Egyptian city of Syene, which indicated that the Sun was directly overhead at that time. He also knew the Sun was not directly overhead at that time where he lived (Alexandria, in northern Egypt) but was about 1/50 of a circle (about 7.2 ) below the zenith. Since Alexandria was about 550 miles north of Syene, then Eratosthenes estimated that the Earth s circumference was about = miles, a size that is only about 10% higher than the actual circumference. A distance that required more mathematically sophisticated methods was derived during the 2 nd century BCE by the Greek mathematician Hipparchus. He applied his newly developed sin(dab) R = sin(adb). AB Also, AB is a chord of the angle ACB, and so AB = 2r sin(cab/2). These results give, R r = 2 sin(51) sin(5) sin(.111) = The Earth-Moon distance is usually expressed as the distance between their centers, and so these calculations give this distance as approximately 71 times the radius of the Earth. The actual distance is about 60 Earth radii, so this was a reasonably good approximation given the limitations of the observations. Note. Hipparchus did not know about sines, but he did know about chords and the law of chords for oblique triangles that is 1

2 Solar System Distances: the Sun A more difficult problem was to determine the Earth-Sun distance, called the Astronomical Unit (AU). It is possible to derive distances to other planets in terms of the AU using the same methods Hipparchus used, but their actual distances could not be derived until the AU was known. The Parallax Method can t be used directly to obtain AU because the Sun is so bright it was not possible to observe any distant objects, such as other stars, behind the Sun. However, an earlier Greek mathematician, Aristarchus ( BCE) realized that when exactly half of the Moon is illuminated as seen from Earth, then the angle formed by the Earth-Moon-Sun must be a right angle. By measuring the apparent angle between the Moon and the Sun, Moon-Earth-Sun, he could express the distance to the Sun, ES, in terms of the distance to the Moon, EM. Aristarchus measured this angle to be 87, and so the Sun s distance relative to the Moon s distance based on this value is, equivalent to the law of sines. Also, he would have used the approximation, ES 1 = = 19. EM cos(87) Unfortunately, his value for the Moon-Earth-Sun angle was not very accurate, and so ES = 19(EM ) is considerably short of the actual distance, ES = 400(EM ). Furthermore, it was difficult for him to determine precisely when the Moon was exactly halfilluminated. A more accurate determination of AU would have to wait until better instruments were developed. Nevertheless, it was a remarkable and audacious achievement of these men to first ask the questions, how distant are the Moon and Sun, and then to apply valid mathematical methods in their attempts to answer those questions. Chord(θ) = 2 sin(θ/2) θ, when θ is small for the very small angle ADB and with θ expressed in radians. Note. If we combine the value Eratosthenes obtained for the Earth s circumference and the value Hipparchus used for π, π (60)(180) = , 3438 then the distance of the Moon from Earth s center is R (70.9)(27500)(3438) = 310, 000. (60)(360) The actual mean distance is about 240,000 miles. In addition, we can derive the Moon s diameter from these results. Since the Moon has the same angular size as the Sun, (0.554 ), then the diameter of the Moon can be expressed as a chord of this angle. Hence, these results give Dm (69.9)(.554)(27500) = 2958, 360 compared to the actual diameter of the Moon which is 2159 miles. The methods used by Hipparchus were correct. The main source of error in his computations was the imprecise value given to the angle ADB. This angle, which represents the angular change in position of an object relative to a very distant background seen from two different vantage points, is called the parallax of the object. The method Hipparchus used to derive the moon s distance is called the Parallax Method. It is still used today. Figure 2: Parallax Method The Parallax Method of distance determination is based on the simple relationship between the angular change in the position of an object relative to a very distant background when viewed from different vantage points and the distance between those vantage points. This is illustrated in Figure 2. An object at C is observed at points A and B. The apparent shift in position 2

3 of C against a distance background is measured by the angle θ. The distance AD is called the baseline. This method is limited by our ability to measure very small angles, and so to determine the distance to very far objects, we must use as long a baseline as possible. Astronomers realized that the longest possible baseline that could be used to measure astronomical distances is obtained by measuring positions of objects when the Earth is at opposite points in its orbit around the Sun. That is, if the Sun is at D and C is observed at two times, A, B, that are six months apart, then the baseline for parallax measurements would be AU. Of course, this requires an accurate determination of AU. When Galileo constructed his telescope and discovered that Venus has phases like the Moon, and that Jupiter has Moons of its own orbiting it like a miniature solar system, the idea of a universe with our Earth at its center was completely and irrevocably shattered. It also became clear that the telescope could help overcome the problems encountered by Greek astronomers by providing more precise measurements of stellar positions and parallax angles. This new instrument, combined with the mathematical tools of trigonometry and the laws of planetary motion discovered by Johannes Kepler and eventually justified by Isaac Newton s Theory of Gravity provided new opportunities to measure our Solar System and beyond. The Italian-French astronomer Giovanni Domenico Cassini, recognized that an upcoming opposition of Mars in 1672 would provide an opportunity to accurately measure the parallax of Mars, which in turn could be used to obtain the Sun s parallax. When Mars is at opposition, that is, when it is directly opposite the Sun as seen from Earth, it comes closest to Earth, and so its parallax is larger than at any other time in its orbit. To provide a large baseline from which to obtain this parallax, Cassini sent a young French astronomer, Jean Richer, to Cayenne in South America. Richer was assigned to measure the position of Mars relative to nearby stars from Cayenne at the exact time of opposition while Cassini did the same from Paris. Once Richer returned to Paris, Cassini combined the two data sets and determined that the parallax of Mars was 25 arcseconds. Since the distance between Cayenne and Paris is about 4000 miles, then Cassini computed the distance between Earth and Mars at the time of these measurements to be = 33 million miles. 25 π Kepler s 1 st law of planetary motion states that planetary orbits are ellipses and his 3 rd law states that the square of a planet s orbital period is directly proportional to the cube of the semimajor axis of its orbit. That is, can vary significantly, depending on where the two planets are in their orbits relative to their aphelion and perihelion positions. In the case of the 1672 opposition, Mars was close to its perihelion (closest to the Sun) and Earth was close to its aphelion (furthest from the Sun). Cassini determined that Mars was approximately 0.38 AU from Earth at that time. He then derived a value for AU of 33/0.38 = 86.8 million miles, which is very close to the accepted mean distance of 93 million miles. That was staggering news at the time, for it expanded the size of our Solar System by twenty-fold over the value obtained by Aristarchus. Kepler s 3 rd law gives us the ability to obtain the distance of any planet to the Sun in terms of Earth s distance, the AU, from that planet s orbital period, its year. By obtaining an accurate value of AU, Cassini unlocked the distances and sizes of other planets in the solar system. This gave us, for the first time in the history of our species, an accurate understanding of the scale of our Solar System. For example, using the modern value for the average angular diameter of the Sun of 0.533, we find that the Sun s diameter is 865,000 miles (see Figure 2). Jupiter s orbital period is years, and so is /3 = 5.2 AU from the Sun. That corresponds to 483 million miles. Knowledge of its distance enables us to derive Jupiter s diameter from its angular size. This gives an equatorial diameter of 88,700 miles, about 11 times the Earth s diameter. Saturn is almost twice Jupiter s distance from the Sun at 886 million miles, with an equatorial diameter of 74,900 miles. The scale of our Solar System is difficult to comprehend, and yet the center of it, our Sun, is just one of what appear to be uncountably many stars in the sky. How far away are they? ( Pm ) 2 = ( Rm ) 3. P e R e However, because Mars has a much greater orbital eccentricity than Earth, the distance between Mars and Earth at opposition 3

4 Distances to the Stars and Beyond Yerkes Observatory also pioneered the use of photography for the determination of stellar parallax. This produced a 10-fold increase in the accuracy of these measurements and significantly extended the size of the known Universe. The new telescopes that were built in the late 19th and early 20th century made many other significant discoveries. They also contributed to a great mystery that puzzled astronomers of this time. As telescopes were improved and made larger, and as finelymachined measurement instruments were attached to these telescopes, it became possible to obtain parallaxes of nearby stars using a baseline that stretched across opposite positions in Earth s orbit (see Figure 2). To illustrate the very small angles these instruments were required to measure, imagine drawing a circle, and then partitioning it into 360 equal arcs. The angular size of each arc is 1. Now partition each of those arcs into 60 equal subarcs. The angular size of these subarcs is 1 arcminute. Further subdivide each of those into 60 equal subarcs and you now have an arc whose angular size is 1 arcsecond. So an arcsecond is 1/3600 of a degree, and there are 1,296,000 arcseconds in a full circle. Astronomers define a parsec to be the distance of an object that has a parallax of 1 arcsecond relative to a baseline width of one AU. Since the sine and tangent of a very small angle are essentially the same as the angle expressed in radians, then 1 parsec corresponds to a distance in miles of 9.3E = π Light travels 186,000 miles per second in the vacuum of space, and so the time it takes light to travel 1 parsec is seconds, or 3.26 years. That is, one parsec is equivalent to a distance of 3.26 light-years. The challenge to astronomers is that no star has a parallax as large as 1 arcsecond. Figure 3: M20, The Trifid Nebula in Sagittarius It was not until that telescopes and parallaxmeasuring instruments became sufficiently powerful and accurate to permit the determination of a star s parallax. Friedrich Struve published his measurement of the parallax of Vega, the brightest star in the constellation Lyra, as arcseconds (26 light-years) in 1837, but the reliability of his measurement was poor, even though his first published value is very close to the currently accepted value of arcseconds. In 1838 Friedrich Wilhelm Bessel obtained a very accurate parallax of arcseconds (10.4 light-years) for the star 61 Cygni. Shortly thereafter in 1839, Thomas Henderson published his parallax of Alpha Centauri, 0.75 arcseconds (4.3 light-years). These successful measurements of stellar parallaxes encouraged observatories, with funding from wealthy philanthropists, to build increasingly larger telescopes that could extend our knowledge of the scale of the Universe. Four such large telescopes were built in the United States. The University of California s Lick Observatory added its 3600 refractor in 1889, and in 1897 the University of Chicago s Yerkes Observatory completed its 4000 refractor. In 1908 Mt. Wilson Observatory completed its 6000 reflector, at that time the largest in the world. It later added the Hooker reflector in These classically beautiful telescopes were able to obtain many more stellar parallaxes. Figure 4: M13, Globular Cluster in Hercules On late fall and early winter evenings, high overhead in the constellation Andromeda, there is a small fuzzy patch in the sky that can be seen with the unaided eye away from city lights. This object, known as M31 from Charles Messier s catalog, is one of largest examples of what were referred to as spiral nebulae by 4

5 late 19th and early 20th century astronomers due to their noticeable spiral structure. As the new telescopes at Lick, Yerkes, and Mt. Wilson began close scrutiny of Spiral Nebulae, some astronomers started getting hints that these objects were not just clouds of gas, but instead contained stars. Astronomers were familiar with open star clusters such as the Pleiades, globular clusters such as M13 in Hercules, and ordinary Gaseous Nebulae such as M20, the Trifid Nebula, in Sagittarius. But Spiral Nebulae were very different from other objects and their sizes were unknown because their distances were unknown. What was the nature of these objects and how distant were they? wavelengths of light that were emitted by various chemical elements, thus proving that Fraunhofer lines represent absorption of light from the Sun by these elements (see Figure 6). This was the beginning of a new science, called spectroscopy, that enabled astronomers to identify the chemical composition of stars without obtaining a physical sample from them. However, the reason atoms only absorbed or emitted specific wavelengths of light was not yet known. James Clerk Maxwell published his equations that characterize electromagnetic fields in a series of papers beginning in Classical Electromagnetic theory is based on these equations and they imply that light is an electromagnetic wave. But it was difficult for scientists of that time to understand how the energy of an electromagnetic wave could move across the space between stars without something that is waved, like an ocean that carries the energy of ocean waves. Figure 6: Light from the Sun is passed through a prism to spread out its colors. Fraunhofer lines are the black lines and represent missing wavelengths of light that have been absorbed by atoms before reaching Earth. Figure 5: M31, Spiral Nebula in Andromeda In 1905, Albert Einstein published three remarkable papers. The first paper, on the photoelectric effect, provided a basis for Quantum theory. The second, his Special Theory of Relativity, describes the nature of space and time and the strange behavior of matter when it travels close to the speed of light. His third paper solved the problem of Brownian motion and proved the existence of atom-sized molecules. In 1913, Niels Bohr introduced his model for atoms which was later refined by Quantum Mechanics and which explained Fraunhofer lines. In 1915, Einstein published his General Theory of Relativity which was verified in spectacular fashion by the 1919 Solar Eclipse. Sir Arthur Eddington s expedition to the island of Prı ncipe to photograph stars near the Sun during totality proved that their apparent positions had been perturbed by amounts correctly predicted by the General Theory. Einstein s description of the equivalence of mass and energy, expressed by the iconic equation E = mc2, would lead eventually to understanding how stars are fueled. Unfortunately, there is a physical limit in our ability to measure stellar parallaxes. The wave nature of light and the optics of telescopes cause these point sources to appear as disks, called Airy disks, due to diffraction. Earth-bound telescopes are further limited by atmospheric distortions that smear stellar images and cause them to move around under high magnification, much like the shimmering air above an asphalt highway on a hot day. So questions about the nature of spiral nebulae sat unanswered, waiting for a great leap in our ability to measure the universe. That leap would come from an unlikely source. Henrietta Leavitt and Cepheid Variables At the beginning of the 20th century, Astronomy, along with the rest of Science, was on the cusp of a major revolution in our understanding of the Universe and how it functions. Earlier in 1814, Joseph von Fraunhofer had closely examined dark lines that appeared in Solar spectra and carefully measured their wavelengths. Then in 1859, Gustav Kirchhoff and Robert Bunsen showed that these dark lines were located at the same The Universe, however, was assumed to be static with some individual objects, such as comets, novae and variable stars, changing, but with a basic structure that did not change. Our Universe was referred to as the Milky Way after the faint band of light that is best seen in summer from a dark location. As5

6 Figure 7: Large and Small Magellanic Clouds with the Milky Way at Cerro Tololo Observatory in Chile tronomers could see that the Milky Way contained a variety of different objects, but most of their distances were unknown. Although spectroscopy could determine the chemical composition of stars and ordinary nebulae, with no means to measure distances beyond a few hundred light-years from Earth, the true nature of our Universe was unknowable and unimaginable. There is a reason why all of the names mentioned thus far are the names of men. Relatively few women attended college then, and those who did had very limited opportunities to apply their knowledge and skills other than by teaching or nursing. One exception to this was provided by Harvard Observatory. Edward Charles Pickering was appointed director of the observatory in 1877 and served in that position until his death in As photography began to replace visual observation in the late 19 th century, data in the form of photographic plates was accumulating much faster than it could be catalogued and analyzed. Pickering recognized that a hitherto untapped resource consisting of intelligent, well-educated women, could provide valuable assistance in the tedious work of cataloguing this ever-expanding collection of astronomical data and by performing related computations. Of course, it is likely that the success of his arguments to the Harvard Corporation to hire women was enhanced by the fact that women could be hired at a considerably lower salary than men, and that these women would not balk at being assigned to perform tiresome, uninteresting duties associated with data-reduction and cataloguing. One of these women was Henrietta Leavitt, born in 1868 and educated at what is now known as Radcliffe College. Her family was sufficiently wealthy to afford her education and Miss Leavitt did not need to work to support herself. After graduation, she traveled around the country and Europe, but became ill and eventually lost her hearing. She had fallen in love with Astronomy during her senior year of college and so, after returning home to Cambridge, Massachusetts, she began volunteer work at Harvard Observatory in Seven years later she was hired by Pickering at a salary of $0.30/hour. That salary was less than what Pickering paid his secretary. Pickering gave Miss Leavitt two principal assignments. The first was to develop a method to derive stellar magnitudes (apparent brightness) from photographic plates, and the second was to identify and catalogue variable stars. To complete the first assignment, she devised and then applied new analytical tools to identify stars near the North Celestial Pole that comprised a sequence of standard magnitudes for comparison with other stars in their vicinity. She then extended this work to identify standard sequences in other sections of the sky. Her standards were utilized until technology advanced sufficiently to permit photometric measurements. This work facilitated her search for variable stars and she discovered over 2400 variables. Henrietta Leavitt possessed an inquiring intelligence that was not satisfied with simply transferring information from photographic plates to a written catalogue. She also began to see patterns in this data and wanted to understand their meaning. One particular set of plates she analyzed came from Harvard s southern hemisphere Observing Station and contained photographic surveys taken over 13 years of the Small and Large Magellanic Clouds. The SMC and LMC are two clusters of stars that can be seen only from the southern hemisphere or low northern latitudes. Armed with her newly developed methods for magnitude determination from photographic plates, Leavitt was able to identify in the SMC 25 members of a type of variable star now called a Cepheid variable. A Cepheid variable changes brightness in a periodic way that can be recognized by its light curve, the relationship between its brightness and time. Since the Cepheids Henrietta Leavitt had discovered were located in the SMC, she correctly reasoned that they were all essentially the same distance from Earth. This implied to her that the differences in brightness among these stars represented differences in their intrinsic brightness, or luminosity. Her curiosity led her to plot the period of these Cepheids versus their brightness. Figure 8 displays her data. As Miss Leavitt wrote in the 1912 paper that announced her discovery, A remarkable relation between the brightness of these variables and the length of their periods will be noticed. Why was this relationship important to Astronomy? Imagine having a light bulb and a meter that can measure light intensity. We know from the inverse square law that the energy of light decreases with the square of the distance from the light source. Suppose we use the meter to measure the apparent brightness of this bulb at a distance accurately measured to be 6

7 Figure 8: Henrietta Leavitt s Data Figure 9: M51, the Whirlpool Galaxy 100 feet and obtain a brightness value of 20. Now suppose we move the bulb to a different location, aim the meter at the bulb, and obtain a reading of 5 for its apparent brightness. Since this value is 1/4 of the brightness at 100 feet, then the distance must be = 200 feet. Comparing the luminosity of a star to its apparent brightness gives distance to the star in a similar way. The relationship between period and luminosity discovered by Henrietta Leavitt made possible the ability to determine luminosity of Cepheids from their periods. A comparison of the luminosity of a Cepheid to its apparent brightness could give its distance. It remained to calibrate the Period-Luminosity relationship of Cepheid variables. As noted by Miss Leavitt in her paper, It is to be hoped, also, that the parallaxes of some variables of this type may be measured. ble from the Cepheid P-L relationship proved conclusively that this object is far outside the Milky Way, about 2.5 million light years away. Astronomers now understood that Spiral Nebulae were in fact Spiral Galaxies, separate and very distant from our own Milky Way Galaxy. Hubble then identified Cepheid variables in other galaxies, thus providing him with their distances. He combined this information with spectroscopic data and noticed that the Fraunhofer lines of galaxies were shifted toward the red by amounts that were directly related to their distances. This galactic red-shift provided the first evidence for the expansion of our universe from a Big Bang beginning, now known to have occurred about 13.7 billion years ago. Distances determined by Cepheid variables enabled astronomers to calibrate other yardsticks, such as Type-1a supernovae, that greatly extended our ability to measure the universe. These yardsticks have produced new challenges for astronomers in their quest to understand the nature of our universe. Leavitt was not free to pursue her own interests at the Observatory, and so she continued her work to catalogue information recorded by other astronomers. One year after her announcement, Ejnar Hertzsprung obtained the parallaxes of several Cepheid variables in the Milky Way, thus providing an initial calibration of the Period-Luminosity relationship of Cepheid variables. Harlow Shapley extended this work by measuring the parallaxes of a much larger number of Cepheids to recalibrate the P-L relationship. He then used his recalibrated yardstick to measure distances to globular clusters that surround the Milky Way. His work put limits on the size of the Milky Way. After Pickering s death, Shapely was named director of the Harvard Observatory in 1921 and then promoted Miss Leavitt to be head of Stellar Photometry. Later that year she died of cancer. Henrietta Leavitt and the other women who worked at Harvard Observatory made significant contributions to Astronomy in spite of strong cultural prejudices of that time against the idea of women gaining advanced academic degrees and holding professional positions. Their successes helped pave the way for other women to enter previously male-dominated scientific fields and confirmed that society benefits most when it allows full use of all of its collective intelligence. Under the leadership of Harlow Shapely, Harvard Observatory created a fellowship that enabled women to study Astronomy at the Observatory. One of the first recipients of that fellowship later became the first woman to chair an academic department at Harvard. The past 100 years have seen spectacular advances in our understanding of the universe and how it functions, much of which stands on the shoulders of these extraordinary women. Sadly, Henrietta Leavitt did not live long enough to witness the greatest triumphs of her discovery. In 1925, Edwin Hubble announced his identification of Cepheid variables in the Andromeda Spiral Nebula using photographs taken with the Hooker telescope at Mt. Wilson. The distance derived by Hub7

8 Further reading The 1912 paper announcing Henrietta Leavitt s discovery of the Period-Luminosity relationship of Cepheid variables can be found here. Note that the author is listed as Edward C. Pickering. Because Miss Leavitt was not a member of the Observatory faculty, she was not permitted to submit papers to the Circular. However, Pickering immediately gives credit to Miss Leavitt in the paper. More information about Henrietta Leavitt and the other women who worked at Harvard Observatory during this time can be found here. 1. Johnson, George Miss Leavitt s Stars: The Untold Story of the Woman Who Discovered How to Measure the Universe. W.W. Norton, London computers.htm Figure 10: M104, the Sombrero Galaxy 3. variable cepheids.html The following references give more detailed technical information regarding the Period-Luminosity relationship of Cepheid variables and its calibration. 1. Koen, C., Kanbur, S. and Ngeow, C The detailed forms of the LMC Cepheid PL and PLC relations. Mon. Not. R. Astron. Soc. 380, Madore, B.F. and Freedman, W.L Calibration of the Extragalactic Distance Scale. Stellar Astrophysics for the Local Group: VIII Canary Islands Winter School of Astrophysics. Ed. A. Aparicio, A. Herrero, and f. Sanchez. Cambridge University Press. Last, but not least, you can find new images of our Universe every day at the Astronomy Picture of the day: Many of the images included here were displayed at APOD. Figure 11: Henrietta Leavitt 8